Rethinking the Flow-based Gradual Domain Adaptation: A Semi-Dual Optimal Transport Perspective

Gradual domain adaptation (GDA) aims to mitigate domain shift by progressively adapting models from the source domain to the target domain via intermediate domains. However, real intermediate domains are often unavailable or ineffective, necessitating the synthesis of intermediate samples. Flow-based models have recently been used for this purpose by interpolating between source and target distributions. Notably, their training typically relies on sample-based log-likelihood estimation, which can discard useful information and thus degrade GDA performance. The key to addressing this limitation is constructing the intermediate domains via samples directly. To this end, we propose an $\underline{\text{E}}$ntropy-regularized $\underline{\text{S}}$emi-dual $\underline{\text{U}}$nbalanced $\underline{\text{O}}$ptimal $\underline{\text{T}}$ransport (E-SUOT) framework to construct intermediate domains. Specifically, we reformulate flow-based GDA as a Lagrangian dual problem and derive an equivalent semi-dual objective that circumvents the need for likelihood estimation. However, the dual problem leads to an unstable min–max training procedure. To alleviate this issue, we further introduce the entropy regularization to convert it into a more stable sequential optimization procedure. Based on this, we propose a novel GDA training framework and provide theoretical analysis in terms of stability and generalization. Finally, extensive experiments are conducted to demonstrate the efficacy of the E-SUOT framework.

Machine learning models often perform poorly when they are used in a new environment that looks different from the data they were trained on, such as applying a model trained on one image style, user group, or language setting to another. A common way to reduce this gap is to adapt the model gradually through intermediate steps, but in many real applications these intermediate data are unavailable and must be created artificially. Existing flow-based methods try to create such intermediate data by estimating the full probability pattern of the target data, which may be inaccurate and may mislead the adaptation process. We propose E-SUOT, a new method that moves source samples toward the target domain more directly, without needing to estimate the target data distribution explicitly. Our approach uses ideas from optimal transport, which can be understood as finding an efficient way to reshape one collection of samples into another, and adds a stabilizing term to make training more reliable. Experiments show that E-SUOT improves target-domain accuracy compared with several existing methods.